Research ArticleNo Access

An improvement of the lower bound for the minimum number of link colorings by quandles

H. Abchir

https://orcid.org/0000-0001-5644-2770

Hassan II University, Casablanca, Morocco

E-mail Address: hamid.abchir@univh2c.ma

Search for more papers by this author

and

S. Lamsifer

Hassan II University, Casablanca, Morocco

E-mail Address: soukaina.lamsifer-etu@etu.univh2c.ma

Search for more papers by this author

https://doi.org/10.1142/S021821652350044XCited by:0 (Source: Crossref)

Abstract

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree $k$ $k$ of the reduced Alexander polynomial of the knot. We show that it is exactly $k + 1$ $k + 1$ for L-space knots. Then we apply these results to torus knots and Pretzel knots $P (- 2, 3, 2 l + 1)$ $P (- 2, 3, 2 l + 1)$ , $l \geq 0$ $l \geq 0$ . We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more than one component.

Keywords:

AMSC: 57K10, 57K12

References

1. H. Abchir, M. Elhamdadi and S. Lamsifer, On the minimum number of Fox colorings of knots, Grad. J. Math. 5 (2020) 122–137. Google Scholar
2. M. R. Adhikari and A. Adhikari, Basic Modern Algebra with Applications (Springer, India, 2014). Crossref, Google Scholar
3. K. E. L. Agle, Alexander and Conway polynomials of torus knots, Master’s Thesis, University of Tennessee (2012). Google Scholar
4. Y. Bae, Coloring link diagrams by Alexander quandles, J. Knot Theory Ramifications 21(10) (2012) 13 pp. Link, Web of Science, Google Scholar
5. F. Bento and P. Lopes, The minimum number of Fox colors modulo 13 is 5, Topol. Appl. 216 (2017) 85–115. Crossref, Web of Science, Google Scholar
6. M. Elhamdadi and J. Kerr, Fox coloring and the minimum number of colors, Involve 10(2) (2016) 291–316. Crossref, Google Scholar
7. R. H. Fox, A quick trip through knot theory, in Topology of 3-Manifolds and Related Topics (Prentice-Hall, 1962). Google Scholar
8. J. Ge, X. Jin, L. H. Kauffman, P. Lopes and L. Zhang, Minimal sufficient sets of colors and minimum number of colors, J. Knot Theory Ramifications 26 (2017) 16 pp. Link, Web of Science, Google Scholar
9. Y. Han and B. Zhou, The minimum number of colorings of knots, J. Knot Theory Ramifications 31(2) (2022) 55 pp. Link, Google Scholar
10. E. Hironaka, The Lehmer polynomial and Pretzel links, Canad. Math. Bull. 44(4) (2001) 440–451. Crossref, Web of Science, Google Scholar
11. K. Ichihara and E. Matsudo, A lower bound on minimal number of colors for links, Kobe J. Math. 33 (2016) 53–60. Google Scholar
12. L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258. Crossref, Web of Science, Google Scholar
13. A. Kawauchi, A Survey of Knot Theory (Birkhäuser Verlag, 1996). Google Scholar
14. W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics, Vol. 175 (Springer, 1997). Crossref, Google Scholar
15. T. Lidman and A. H. Moore, Pretzel knots with L-space-surgeries, Michigan Math. J. 65 (2016) 105–130. Crossref, Web of Science, Google Scholar
16. T. W. Mattman and P. Solis, A proof of the Kauffman–Harary conjecture, Algebr. Geom. Topol. 9(4) (2009) 2027–2039. Crossref, Web of Science, Google Scholar
17. K. Murasugi, On the braid index of alternating links, Trans. Amer. Math. Soc. 326(1) (1991) 237–260. Crossref, Web of Science, Google Scholar
18. T. Nakamura, Y. Nakanishi and S. Satoh, The pallet graph of a Fox coloring, Yokohama Math. J. 59 (2013) 91–97. Google Scholar
19. T. Nakamura, Y. Nakanishi and S. Satoh, On effective 9-colorings for knots, J. Knot Theory Ramifications 23(12) (2014) 1450059. Link, Web of Science, Google Scholar
20. T. Nakamura, Y. Nakanishi and S. Satoh, $11$ $11$ -colored knot diagram with five colors, J. Knot Theory Ramifications 25(4) (2016) 22 pp. Link, Web of Science, Google Scholar
21. S. Nelson, Classification of finite Alexander quandles, in Proc. Spring Topology and Dynamical System Conf., Topology Proceedings, Vol. 27, No. 1, 2003, pp. 245–258. Google Scholar
22. K. Oshiro, Any 7-colorable knot can be colored by four colors, J. Math. Soc. Japan 62(3) (2010) 963–973. Crossref, Web of Science, Google Scholar
23. P. Ozsváth, A. Stipsicz and Z. Szabó, Concordance homomorphisms from knot Floer homology, Adv. Math. 315 (2017) 366–426. Crossref, Web of Science, Google Scholar
24. P. Ozsváth and Z. Szabó, On knot Floer homology and lens space surgeries, Topology 44(6) (2005) 1281–1300. Crossref, Web of Science, Google Scholar
25. S. Satoh, 5-colored knot diagram with four colors, Osaka J. Math. 46(4) (2009) 939–948. Web of Science, Google Scholar